Theorem splcl 11736

Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)

Assertion

Ref	Expression
splcl	|- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) e. Word A )

Proof of Theorem splcl

Dummy variables s b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2924	. . . 4 |- ( S e. Word A -> S e. _V )
2		otex 4388	. . . 4 |- <. F , T , R >. e. _V
3		id 20	. . . . . . . 8 |- ( s = S -> s = S )
4		fveq2 5687	. . . . . . . . . 10 |- ( b = <. F , T , R >. -> ( 1st ` b ) = ( 1st ` <. F , T , R >. ) )
5	4	fveq2d 5691	. . . . . . . . 9 |- ( b = <. F , T , R >. -> ( 1st ` ( 1st ` b ) ) = ( 1st ` ( 1st ` <. F , T , R >. ) ) )
6	5	opeq2d 3951	. . . . . . . 8 |- ( b = <. F , T , R >. -> <. 0 , ( 1st ` ( 1st ` b ) ) >. = <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. )
7	3, 6	oveqan12d 6059	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) = ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) )
8		simpr 448	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> b = <. F , T , R >. )
9	8	fveq2d 5691	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 2nd ` b ) = ( 2nd ` <. F , T , R >. ) )
10	7, 9	oveq12d 6058	. . . . . 6 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) = ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) )
11		simpl 444	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> s = S )
12	8	fveq2d 5691	. . . . . . . . 9 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 1st ` b ) = ( 1st ` <. F , T , R >. ) )
13	12	fveq2d 5691	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 2nd ` ( 1st ` b ) ) = ( 2nd ` ( 1st ` <. F , T , R >. ) ) )
14	11	fveq2d 5691	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( # ` s ) = ( # ` S ) )
15	13, 14	opeq12d 3952	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. = <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. )
16	11, 15	oveq12d 6058	. . . . . 6 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) = ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) )
17	10, 16	oveq12d 6058	. . . . 5 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) concat ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
18		df-splice 11682	. . . . 5 |- splice = ( s e. _V , b e. _V |-> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) concat ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) )
19		ovex 6065	. . . . 5 |- ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. _V
20	17, 18, 19	ovmpt2a 6163	. . . 4 |- ( ( S e. _V /\ <. F , T , R >. e. _V ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
21	1, 2, 20	sylancl 644	. . 3 |- ( S e. Word A -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
22	21	adantr 452	. 2 |- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
23		swrdcl 11721	. . . . 5 |- ( S e. Word A -> ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A )
24	23	adantr 452	. . . 4 |- ( ( S e. Word A /\ R e. Word A ) -> ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A )
25		ot3rdg 6322	. . . . . 6 |- ( R e. Word A -> ( 2nd ` <. F , T , R >. ) = R )
26	25	adantl 453	. . . . 5 |- ( ( S e. Word A /\ R e. Word A ) -> ( 2nd ` <. F , T , R >. ) = R )
27		simpr 448	. . . . 5 |- ( ( S e. Word A /\ R e. Word A ) -> R e. Word A )
28	26, 27	eqeltrd 2478	. . . 4 |- ( ( S e. Word A /\ R e. Word A ) -> ( 2nd ` <. F , T , R >. ) e. Word A )
29		ccatcl 11698	. . . 4 |- ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A /\ ( 2nd ` <. F , T , R >. ) e. Word A ) -> ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A )
30	24, 28, 29	syl2anc 643	. . 3 |- ( ( S e. Word A /\ R e. Word A ) -> ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A )
31		swrdcl 11721	. . . 4 |- ( S e. Word A -> ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A )
32	31	adantr 452	. . 3 |- ( ( S e. Word A /\ R e. Word A ) -> ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A )
33		ccatcl 11698	. . 3 |- ( ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A /\ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A ) -> ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. Word A )
34	30, 32, 33	syl2anc 643	. 2 |- ( ( S e. Word A /\ R e. Word A ) -> ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. Word A )
35	22, 34	eqeltrd 2478	1 |- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) e. Word A )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   <.cop 3777   <.cotp 3778   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   0cc0 8946   #chash 11573  Word cword 11672   concat cconcat 11673   substr csubstr 11675   splice csplice 11676

This theorem is referenced by:  efglem  15303  efgtf  15309  frgpuplem  15359  psgnunilem2  27286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-concat 11679  df-substr 11681  df-splice 11682